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On Weakly Complete Group Algebras of Compact Groups

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On Weakly Complete Group Algebras of Compact Groups Journal of Lie Theory Volume 30 (2020) 407–424 © 2020 Heldermann Verlag On Weakly Complete Group Algebras of Compact Groups Karl Heinrich Hofmann and Linus Kramer∗ Communicated by K.-H. Neeb Abstract. A topological vector space over the real or complex field K is weakly complete if it is isomorphic to a power KJ . For each topological group G there is a weakly complete topological group Hopf algebra K[G] over K = R or C, for which three insights are contributed: Firstly, there is a comprehensive structure theorem saying that the topological algebra K[G] is the cartesian product of its finite dimensional minimal ideals whose structure is clarified. Secondly, for a compact abelian group G and its character group Gb , the weakly complete complex b b b× b ∼ b b Hopf algebra C[G] is the product algebra CG with the comultiplication c: CG ! CG G = CG ⊗CG , b Gb Gb c(F )(χ1; χ2) = F (χ1 + χ2) for F : G ! C in C . The subgroup Γ(C ) of grouplike elements of b the group of units of the algebra CG is Hom(G;b (C n f0g;:)) while the vector subspace of primitive ∼ b ∼ elements is Hom(G;b (C; +)). This forces the group Γ(R[G]) ⊆ Γ(C[G]) to be Hom(G;b S1) = Gb = G ∼ with the complex circle group S1 . While the relation Γ(R[G]) = G remains true for any compact ∼ group, Γ(C[G]) = G holds for a compact abelian group G if and only if it is profinite. Thirdly, for each pro-Lie algebra L a weakly complete universal enveloping Hopf algebra UK(L) over K exists such that for each connected compact group G the weakly complete real group Hopf algebra R[G] is a quotient Hopf algebra of UR(L(G)) with the (pro-)Lie algebra L(G) of G. The group Γ(UR(L(G))) of grouplike elements of the weakly complete enveloping algebra of L(G) ∼ maps onto Γ(R[G]) = G and is therefore nontrivial in contrast to the case of the discrete classical enveloping Hopf algebra of an abstract Lie algebra. Mathematics Subject Classification: 22E15, 22E65, 22E99. Key Words: Weakly complete vector space, weakly complete algebra, group algebra, Hopf algebra, compact group, Lie algebra, universal enveloping algebra. 1. Introduction For much of the material surrounding a theory of group Hopf algebras in the category of weakly complete real or complex vector spaces we refer to [3] and the forthcoming fourth edition of [6]. Some additional facts are presented here. A topological vector space over a locally compact field K is called weakly complete if it is isomorphic to KJ for some set J . This text considers K = R or K = C only and deals ∗Both authors were supported by Mathematisches Forschungsinstitut Oberwolfach in the pro- gram RiP (Research in Pairs). Linus Kramer is funded by the Deutsche Forschungsgemeinschaft under Germany’s Excellence Strategy EXC 2044-390685587, Mathematics Münster: Dynamics- Geometry-Structure. ISSN 0949–5932 / $2.50 © Heldermann Verlag 408 Hofmann and Kramer with the categories G of topological groups, respectively, WA of weakly complete topological algebras. Inside each WA-object A we have the subset A−1 of all units (i.e., invertible elements) which turns out to be a G -object in the subspace topology. In [3] it was shown that the functor A ! A−1 : WA ! G has a left adjoint functor G 7! K[G]: G ! WA. Automatically, K[G] is a topological Hopf algebra. Then −1 for each topological group G there is a G -morphism ηG : G ! K[G] such that for each WA-object A and G -morphism f : G ! A−1 there is a unique WA-morphism 0 0 f : K[G] ! A such that f(g) = f (ηG(g)). The weakly complete algebra A = K[G] is seen to have a comultiplication c: A ! A ⊗ A making it into a symmetric Hopf algebra. The subset G(A) = fa 2 A−1 : c(a) = a ⊗ ag is a subgroup of A−1 and its elements are called grouplike. If G is a discrete group, then K[G] is the traditional group algebra over K and ηG is an embedding and its image is G(K[G]). In [3] it was shown that for a compact group G the map ηG is an embedding and for K = R induces an isomorphism onto G(R[G]). We shall see in this text that this fails over the complex ground field even for all compact abelian groups which are not profinite. The assignment G 7! G(C[G]) may be viewed as as ‘complexification’ of the compact group G. This viewpoint becomes important if one considers the module category of C[G] in W , i.e. the representations of G on weakly complete complex vector spaces, but we will not pursue this in the present note. Since we shall deal with compact groups throughout this paper, we shall always consider G as a subgroup of the group K[G]−1 of units of K[G]. In particular, this means G ⊆ R[G] ⊆ C[G]. The article [3] and the 4th Edition of the book [6] identify a category H of weakly complete topological Hopf algebras for which the functor G ! K[G] implements an equivalence of the category of compact groups and the category H. The vector space continuous dual of K[G] turns out to be the traditional representation algebra R(G; R). This approach yields a new access to the Tannaka-Hochschild duality of the categories of compact groups and “reduced” real Hopf algebras. In all of this, the precise nature of the weakly complete Hopf algebras K[G] even for compact groups remained somewhat obscure in the nondiscrete case. The present paper will present a precise piece of information on the topological algebra structure of K[G] in terms of a direct product of its finite dimensional minimal ideals whose precise structure links this presentation with the classical information of finite di- mensional G-modules (cf. e.g. [6], Chapters 3 and 4). Certain complications have to be overcome on that level if one insists on an explicit identification of the algebra and ideal structure of R[G] as the case C[G] is easier. For abelian compact groups G we shall present a very direct access to the structure of the weakly complete Hopf algebra of C[G] by identifying an isomorphism between b C[G] and CG and by further identifying the group of grouplike elements to be isomorphic to (L(G); +) ⊕ G with the pro-Lie algebra L(G) of G (see [6]). The subgroup G of C[G]−1 therefore agrees with the group of grouplike elements of C[G] if and only if L(G) = f0g if and only if G is totally disconnected. The presence of weakly complete Lie algebras over K in the group K-Hopf algebra of a compact group motivates a proof of the existence of a weakly complete universal Hofmann and Kramer 409 enveloping algebra over K for WA Lie algebras over K. In contrast to the case of enveloping algebras on the purely algebraic side, the WA enveloping Hopf algebras will sometimes have grouplike elements. The universal property of the WA envelop- ing Hopf algebras will show that each WA Hopf-group algebra K[G] of a compact connected group G is a quotient algebra of the WA enveloping algebra of L(G). So WA enveloping algebras have a tendency of being larger than WA group algebras. For a special class of profinite dimensional Lie algebras a similar but different ap- proach to appropriate enveloping algebras is considered in [4]. A preprint of the present material appeared in the Series of Preprints of the Mathe- matical Research Institute of Oberwolfach [5]. 2. Weakly Complete Hopf Algebras For the basic theory of weakly complete Hopf algebras we may safely refer to [3] and [6], 4th Edition. For the present discussion we need a reminder of some basic concepts. Definition 2.1. Let A be a weakly complete symmetric Hopf algebra, i.e. a group object in the monoidal category (W; ⊗W ) of weakly complete vector spaces (see [6], Appendix 7 and Definition A3.62), with comultiplication c: A ! A ⊗ A and coidentity k : A ! K. An element a 2 A is called grouplike if c(a) = a ⊗ a and k(a) = 1. The subgroup of grouplike elements in the group of units A−1 will be denoted G(A). An element a 2 A is called primitive, if c(a) = a ⊗ 1 + 1 ⊗ a. The Lie algebra of primitive elements of ALie , i.e. the weakly complete Lie algebra obtained by endowing the weakly complete vector space underlying A with the Lie bracket obtained by [a; b] = ab − ba, will be denoted Π(A). We recall that any weakly complete symmetric Hopf algebra A has an exponential ! −1 function expA : A A as explained in [3], Theorem 3.12 or in [6], 4th edition, A7.41. Theorem 2.2. Let A be a weakly complete symmetric Hopf algebra. Then the following statements hold: (i) The set Γ(A) of grouplike elements of a weakly complete symmetric Hopf algebra A is a closed subgroup of (A; ·) and therefore is a pro-Lie group. (ii) The set Π(A) of primitive elements of A is a closed Lie subalgebra of ALie and therefore is a pro-Lie algebra. ∼ G (iii) Π(A) = L( (A)) and the exponential function expA of A induces the exponen- ! tial function expΓ(A) : Π(A) Γ(A) of the pro-Lie group Γ(A).
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